It's not "both" systems, it's at least four. Canada is private doctors, public insurance. In the UK, the public system includes the doctors as well. France is not single-payer but has a system somewhat similar to the ACA, with private but nonprofit insurance, an effective mandate, a national price list for health services, and a prohibition of claim denials for anything on the list.

A great book comparing eight national systems in The Healing of America by T.R. Reid. It references stats showing the French-style systems get the highest quality results, with reasonable cost. Japan and Germany have similar systems.